2019-12-23
A lower bound for point-to-point connection probabilities in critical percolation
Publication
Publication
Consider critical site percolation on Zd with d≥2. We prove a lower bound of order n−d2 for point-to-point connection probabilities, where n is the distance between the points. Most of the work in our proof concerns a `construction' which finally reduces the problem to a topological one. This is then solved by applying a topological fact, which follows from Brouwer's fixed point theorem. Our bound improves the lower bound with exponent 2d(d−1), used by Cerf in 2015 to obtain an upper bound for the so-called two-arm probabilities. Apart from being of interest in itself, our result gives a small improvement of the bound on the two-arm exponent found by Cerf.
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| arXiv.org e-Print archive | |
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van den Berg, R.& Don, H. (2019). A lower bound for point-to-point connection probabilities in critical percolation. In arXiv.org e-Print archive (ArXiv:1912.10964v1). |
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